The objective of our cross-sectional analysis was to investigate that the relationship between bid premiums, duration, P/B, beta, bid-ask spread and spread returns. We modeled this relationship as following:
)
ere i is ith deal offer, SR is spread returns. RR is revision returns; Dur is duration for every deal. BP is bid-premium; P/B is measure as the ratio of total market equity to equity book value; Risk is the idiosyncratic (unsystematic) risk in the market model, and Spread is the bid-ask spread ratio. is the random error
5.1 Model selection
Before going through our model selection, we calculated the variation inflation factors (VIF) of variables to test for the existence of the multicolinearity. We found that the multicolinearity problem of explanatory variables was not serious. The objective of our
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cross-sectional tests was to draw inferences about the relationship between bid premiums, duration, P/B, beta, bid-ask spread and spread returns. We proposed one simpler and two more elaborate techniques. At first, we used ordinary least-squares (OLS) analysis to estimate the relationship between arbitrage returns and variables. This empirical model is both simple and commonly used in other research. Second, we employed a generalized weighted least-squares (GLS) analysis to solve the possible problems of heteroscedasticity and cross-sectional correlations among the residuals7.
Third, an endogeneity problem may have affected our model. We argued that bid premiums, durations and spread returns were jointly determined. Some theoretical models of risk arbitrage indicated that many takeover variables were endogenously determined. These variables include durations, bid premiums and risk arbitrage returns. For example, arbitrageurs were also more likely to increase their holdings when the bid premiums were high (Hsieh & Walkling, 2005). Cornelli & Li (2002) suggested that there was a positive relation between a change of duration and arbitrage returns. One important source of endogeneity is reverse causality of variables. If variables are endogenous, estimates forming ordinary least-squares (OLS) will be biased and inconsistent. Therefore, we employed the endogeneity test to check whether spread returns, duration and bid premiums were exogenous in the equation. The result in Table 8 shows that spread returns and bid premiums are more likely to be endogenous. Therefore, we recognized this potential endogeneity in our analysis, testing the link between spread returns and variables through use of appropriate instruments
7 There is a serious problem in our analysis of OLS. The random errors are not normal distribution in the regression model. As a result, we find the generalized linear model (GLM) is a general framework to deal with non-normal models. GLM is a flexible generalization of ordinary least squares regression. It relates the random distribution of the measured variable of the experiment to the systematic portion of the experiment (the linear predictor) through a link function. Our link function is:
iables t
independen are
X SR of mean is where
X , var
)
ln(μ = β μ
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with two-stage least-squares (2SLS). The requirement for the instrument was that it should be closely correlated with the corresponding dependent variables, but not with other dependent variables. We used a target firm’s size and Run-up as the instrument variables for bid premiums. A target firm’s size is the equity market value at the announcement date, and has positive relation with the bid premium. Besides, Run-up is the cumulative abnormal return to the target firm’s stock for trading days (-30, -1) before the announcement day. Hsieh &
Walkling (2005) reported that Run-up is a good instrument variable for bid premium. In addition, we split the whole sample into successful and failed observations to test again the relationship between spread returns and variables.
5.2 Regression analysis
The relationship between spread returns and variables is shown in equation 3. Within the OLS and GLS frameworks, we obtained the regression results shown in the Table 9. In addition, we split our observations into successful and failed deals, to ensure the effect of spread returns on variables (equation 4). Considering the potential endogenous problem, bid premium is an endogenous variable in equation 3; as a result, we also use 2SLS models (equations 5 and 6) to do our analysis. A targets firm’s size and Run-up are the instrument variables for bid premiums.
)
In equation 4, i is ith deal offer, SR is spread returns. RR is revision returns; Dur is duration for every deal. BP is bid-premium; P/B is measure as the ratio of total market equity to equity book value; Risk is the idiosyncratic (unsystematic) risk in the market model, and Spread is the bid-ask spread ratio. D is dummy variable (D=1, the deal is successful, D=0, the deal is failed). In equations 5 and 6, Size is the market value of target firms; Runup is the cumulative
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abnormal returns to the target firm’s stock for trading days (-30, -1) before the announcement day, BP’ is the predicted values of BP in the equation 4, and X are Dur, P/B, Risk, and Spread, individually (j=0~5, Xo is the intercept).
5.3 The predicting of stock reversal
LBOs are different from general mergers and acquisition: most of them are successful.
There is also a unique attribute in our sample, which some target firms with higher spread returns would result in larger revision returns. The Table 1 shows that most of the observations tend downward. This fact inspired us to investigate the target firms’ stock movement during the deal period. Unlike most previous risk arbitrage research which developed models for predicting acquisition outcomes, we investigated whether the reversal of stock during the period of deals resulted from variables in the deals, and developed a predictive model using logistic regression analysis. We also used the alternative of bid-ask spread ratios with different periods to check whether liquidity could affect the reversal of stock. The model is:
where i is ith deal offer; X is Dur, P/B, Risk, BP and Spread, individually (j=0~5, Xo is the intercept); Dur is the duration for every deal; P/B is price-to-book ratio; Risk is the idiosyncratic (unsystematic) risk in the market model; BP is the bid premium, and Spread is the bid-ask spread ratio.
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